Test instruments and methods for compensating IQ imbalance

ABSTRACT

A test instrument may include a transmitter configured to transmit signals to a unit under test, a receiver configured to receive signals from the unit under test, and a controller configured to generate a transmitter compensation filter by (i) transmitting, with the transmitter, complex multi-sine signals over a first plurality of observed frequencies within a predetermined baseband frequency range, (ii) estimating a first plurality of frequency responses that compensate for in-phase and quadrature (IQ) imbalance at the first plurality of observed frequencies within the predetermined baseband frequency range, and (iii) determining, using the first plurality of frequency responses, a transmitter polynomial surface, and to compensate, using the transmitter compensation filter, at least one of the signals to be transmitted by the transmitter to reduce IQ imbalance in the transmitted signals, including using the transmitter polynomial surface to calculate a frequency response that reduces the IQ imbalance in the transmitted signals.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claim priority to and the benefit of U.S. ProvisionalPatent Application No. 62/691,675, filed Jun. 29, 2018, the entiredisclosure of which is incorporated by reference herein.

TECHNICAL FIELD

The present disclosure relates to test instruments and equipment, andmore specifically, to systems, methods, and devices usingdirect-conversion receivers and transmitters.

BACKGROUND

IQ imbalance is one of the major impairments introduced by analogcomponents (mixer and filters) in direct-conversion (orzero-intermediate frequency (IF)) receivers and transmitters. Thisimpairment produces degradation in the performance of both communicationsystems and test equipment. IQ imbalance is mainly introduced by mixerimperfections and frequency differences between in-phase (I) andquadrature (Q) channels. This impairment produces degradation in theperformance of both communication systems and test equipment.

SUMMARY

According to one aspect of the present disclosure, a test instrument maycomprise a transmitter configured to transmit signals to a unit undertest, a receiver configured to receive signals from the unit under test,and a controller. The controller may be configured to generate atransmitter compensation filter at least by (i) transmitting, with thetransmitter, complex multi-sine signals over a first plurality ofobserved frequencies within a predetermined baseband frequency range,(ii) estimating a first plurality of frequency responses that compensatefor in-phase and quadrature (IQ) imbalance at the first plurality ofobserved frequencies within the predetermined baseband frequency range,and (iii) determining, using the first plurality of frequency responses,a transmitter polynomial surface. The controller may further beconfigured to compensate, using the transmitter compensation filter, atleast one of the signals to be transmitted by the transmitter to reduceIQ imbalance in the transmitted signals, including using the transmitterpolynomial surface to calculate a frequency response that reduces the IQimbalance in the transmitted signals.

In some embodiments, the controller may further be configured togenerate a receiver compensation filter at least by (i) receiving, withthe receiver, complex multi-sine signals over a second plurality ofobserved frequencies within the predetermined baseband frequency range,(ii) estimating a second plurality of frequency responses thatcompensate for IQ imbalance at the second plurality of observedfrequencies within the predetermined baseband frequency range, and (iii)determining, using the second plurality of frequency responses, areceiver polynomial surface. The controller may further be configured tocompensate, using the receiver compensation filter, at least one of thesignals to be received by the receiver to reduce IQ imbalance in thereceived signals, including using the receiver polynomial surface tocalculate a frequency response that reduces the IQ imbalance in thereceived signals.

In some embodiments, to generate the transmission compensation filtermay further comprise transmitting an amplitude imbalance signal inresponse to phases of the transmitter and the receiver not beingsynchronized.

In some embodiments, the controller may be configured to generate thetransmitter compensation filter and the receiver compensation filter atthe same time by offsetting respective operating frequencies of a mixerof the transmitter and a mixer of the receiver during the transmittingand receiving of the complex multi-sine signals.

In some embodiments, the controller may be configured to offset therespective operating frequencies by a multiple of a bin size of a fastFourier transform (FFT) block, such that each of complex sines andimages of the IQ imbalance lie on a bin of the FFT block.

In some embodiments, the first plurality of observed frequencies maycomprise (i) a first observation at a positive predefined frequency whena complex sine signal is transmitted at one of the positive predefinedfrequency and a complex conjugate of the positive predefined frequencyand (ii) a second observation at a negative predefined frequency when acomplex sine signal is transmitted at one of the negative predefinedfrequency and a complex conjugate of the negative predefined frequency.

In some embodiments, the controller may be configured to transmit thecomplex multi-sine signals by transmitting separate complex sine signalsat different times.

In some embodiments, the transmitter polynomial surface may include atleast one frequency outside the predetermined baseband frequency range.

In some embodiments, the IQ imbalance may include IQ imbalance images,and the controller may be configured to reduce the IQ imbalance imagesto be at least 60 decibels (dB) below the signals to be transmitted bythe transmitter.

In some embodiments, the controller may be configured to determine thetransmitter polynomial surface by approximating a multi-variablepolynomial function using a least squares approach.

In some embodiments, the controller may include a transmitter IQimbalance compensator, a receiver IQ imbalance compensator, atransmitter amplitude correction engine, a receiver amplitude correctionengine, a transmitter resampling engine, a receiver resampling engine,an acquisition engine, an arbitration engine, and a trigger routingmatrix, where the acquisition and arbitration engines are both incommunication with the trigger routing matrix.

According to another aspect of the present disclosure, a method maycomprise generating a transmitter compensation filter. Generating thetransmitter compensation filter may comprise the steps of (i)transmitting, with a transmitter, complex multi-sine signals over afirst plurality of observed frequencies within a predetermined basebandfrequency range, (ii) estimating a first plurality of frequencyresponses that compensate for in-phase and quadrature (IQ) imbalance atthe first plurality of observed frequencies within the predeterminedbaseband frequency range, and (iii) determining, using the firstplurality of frequency responses, a transmitter polynomial surface.

In some embodiments, the method may further comprise compensating, usingthe transmitter compensation filter, a signal transmitted by thetransmitter to reduce IQ imbalance in the transmitted signal, includingusing the transmitter polynomial surface to calculate a frequencyresponse that reduces the IQ imbalance in the transmitted signal.

In some embodiments, the method may further comprise generating areceiver compensation filter. Generating the receiver compensationfilter may comprise the steps of (i) receiving, with a receiver, complexmulti-sine signals over a second plurality of observed frequencieswithin the predetermined baseband frequency range, (ii) estimating asecond plurality of frequency responses that compensate for IQ imbalanceat the second plurality of observed frequencies within the predeterminedbaseband frequency range, and (iii) determining, using the secondplurality of frequency responses, a receiver polynomial surface.

In some embodiments, the method may further comprise compensating, usingthe receiver compensation filter, a signal received by the receiver toreduce IQ imbalance in the received signal, including using the receiverpolynomial surface to calculate a frequency response that reduces the IQimbalance in the received signal.

In some embodiments, the transmitter compensation filter and thereceiver compensation filter may be generated at the same time byoffsetting respective operating frequencies of a mixer of thetransmitter and a mixer of the receiver during the transmitting andreceiving of the complex multi-sine signals.

In some embodiments, the first plurality of observed frequencies maycomprise (i) a first observation at a positive predefined frequency whena complex sine signal is transmitted at one of the positive predefinedfrequency and a complex conjugate of the positive predefined frequencyand (ii) a second observation at a negative predefined frequency when acomplex sine signal is transmitted at one of the negative predefinedfrequency and a complex conjugate of the negative predefined frequency.

According to yet another aspect of the present disclosure, a testinstrument may comprise a transmitter configured to transmit signals toa unit under test, a receiver configured to receive signals from theunit under test, and a controller configured to reduce in-phase andquadrature (IQ) imbalance of the signals to be transmitted and received.The controller may be configured to reduce IQ imbalance at least by (i)transmitting complex multi-sine signals over a first plurality ofobserved frequencies within a predetermined baseband frequency range,(ii) receiving the transmitted multi-sine signals over a secondplurality of observed frequencies within the predetermined basebandfrequency range, the second plurality of observed frequencies beingoffset from the first plurality of observed frequencies, (iii)estimating corresponding sets of frequency responses for the transmittedand received signals that compensate for IQ imbalance at the first andsecond pluralities of observed frequencies within the predeterminedbaseband frequency range, (iv) determining, using the sets of frequencyresponses, polynomial surfaces, (v) approximating each of the polynomialsurfaces with a corresponding one of transmitter and receivermulti-variable polynomials, and (vi) reducing the IQ imbalance of thesignals to be transmitted and received based on frequency responsescalculated using the transmitter and receiver multi-variablepolynomials.

In some embodiments, the controller may be configured to reduce the IQimbalance such that IQ imbalance images are at least 60 decibels (dB)below at least one of the transmitted and received signals.

In some embodiments, the controller may be configured to reduce the IQimbalance for frequencies covering a 6-GHz bandwidth.

BRIEF DESCRIPTION OF THE DRAWINGS

The detailed description particularly refers to the following figures,in which:

FIG. 1 is a simplified block view of a testing system that includes atest instrument system and a unit under test (UUT);

FIG. 2 is a schematic block diagram of transmitter circuitry of the testinstrument of FIG. 1;

FIG. 3 is a schematic block diagram of receiver circuitry of the testinstrument of FIG. 1;

FIG. 4 is a simplified block view of a controller of the test instrumentof FIG. 1;

FIG. 5 is a schematic block diagram of the controller logic of the testinstrument of FIG. 1;

FIG. 6 is a table illustrating exemplary parameters that may be used toestimate and compensate for the IQ imbalance;

FIG. 7 is a graph illustrating a multi-sine signal transmitted at 5.125GHz without transmitter pre-compensation received by the receiver ofFIG. 1;

FIG. 8A is a graph illustrating a compensation filter surface of thetransmitter of FIG. 1;

FIG. 8B is a graph illustrating a polynomial surface fit for any RFfrequency based on the estimated compensation filter of FIG. 8A;

FIG. 8C is a graph illustrating a local polynomial surface fit aroundthe RF frequency to compensate that is generated using parameters ofFIG. 6 to estimate and compensate the IQ imbalance;

FIG. 9A is a graph illustrating a compensation filter surface of thereceiver of FIG. 1;

FIG. 9B is a graph illustrating a polynomial surface fit for any RFfrequency based on the estimated compensation filter of FIG. 9A;

FIG. 9C is a graph illustrating a local polynomial fit around a targetedRF frequency that is generated using parameters of FIG. 6 to estimateand compensate the IQ imbalance;

FIG. 10A is a graph illustrating spectrum of images of the TX imbalanceat 5.125 GHz; and

FIGS. 10B and 10C are graphs illustrating output after pre-compensationin the transmitter and post-compensation in the receiver of FIG. 1.

DETAILED DESCRIPTION OF THE DRAWINGS

While the concepts of the present disclosure are susceptible to variousmodifications and alternative forms, specific exemplary embodimentsthereof have been shown by way of example in the drawings and willherein be described in detail. It should be understood, however, thatthere is no intent to limit the concepts of the present disclosure tothe particular forms disclosed, but on the contrary, the intention is tocover all modifications, equivalents, and alternatives falling withinthe spirit and scope of the invention as defined by the appended claims.

Referring now to FIG. 1, a test instrument system 10 configured toevaluate a unit under test (UUT) 24 is shown. In the illustrativeembodiment, the test instrument 10 includes a direct-conversion testsignal generator transmitter 14 configured to transmit test signals tothe UUT 24 and a direct-conversion receiver 16 configured to receivesignals from the UUT 24. As discussed above, IQ imbalance may beintroduced by mixer imperfections and frequency differences betweenin-phase (I) and quadrature (Q) channels. This impairment producesdegradation in the performance of communication systems and in theaccuracy of the test instrument 12. As such, the test instrument 12 isconfigured to compensate the IQ imbalance impairments at the transmitter14 and the receiver 16 to improve the operation of the test instrument12.

As discussed in detail below, the test instrument 12 is configured toestimate compensation filters to remove IQ imbalances for thetransmitter 14 and the receiver 16 at a given RF frequency based onobservations using a compensation algorithm. Specifically, acompensation filter on the transmitter 14 is used to pre-compensate atransmitted signal to the UUT 24 to reduce the IQ imbalance introducedby a transmit (also referred to as TX) analog chain, whereas, acompensation filter on the receiver 16 is used to post-compensate areceived signal from the UUT 24 to reduce the IQ imbalance introduced bya receive (also referred to as RX) analog chain. It should beappreciated that the estimated compensation filters at the observed RFfrequencies may be stored for calibration. In other words, the testinstrument 12 is configured to determine a compensation filter for a setof RF frequencies that may be used to verify whether IQ imbalance imagesare lower than a predefined threshold level. For example, IQ imbalanceimages that are lower than 60 dB below the transmitted/received signalis desired. In the illustrative embodiment, the test instrument 12 isconfigured to compensate for any RF frequency up to 6 GHz by convertingthe observation frequencies to a surface that can be approximated, usinga least squares approach, to a multi-variable polynomial function, asdescribed in greater detail below. It should be appreciated that the IQimbalance compensation process may be implemented on any suitableplatform.

As shown in FIG. 1, the test instrument 12 includes the transmitter orgenerator 14, the receiver 16, a radiofrequency (RF) duplex 18, acontroller 20, and a display 22. The transmitter 14 may be embodied asany type of circuitry and/or components capable of communicating withthe UUT 24 to transmit signals to the UUT 24. The receiver 16 may beembodied as any type of circuitry and/or components capable ofcommunicating signals from the UUT 24 to receive signals from the UUT24. In the illustrative embodiment, the transmitter 14 and the receiver16 communicate through separate ports or connectors with the UUT 24.However, in some embodiments, the transmitter 14 and the receiver 16 maycommunicate with the UUT 24 via the radiofrequency (RF) duplex port orconnector 18. The RF duplex connector 18 may be embodied as any type ofcircuitry and/or components capable of performing bi-directionalcommunication between the transmitter 14/the receiver 16 and the UUT 24by sharing a common connector.

The controller 20 may be embodied as any type of device or collection ofdevices capable of performing various compute functions describedherein. In the illustrative embodiment, the controller 20 is configuredto estimate compensation filters at a given RF frequency for thetransmitter 14 and the receiver 16 based on observations. Additionally,the controller 20 may load the estimated compensation filters to verifythat IQ imbalance images are lower than a predefined threshold.

The display 22 of the test instrument 12 may be embodied as any type ofdisplay capable of displaying digital information such as a liquidcrystal display (LCD), a light emitting diode (LED), a plasma display, acathode ray tube (CRT), or other type of display device. In someembodiments, the display 22 may be coupled to a touch screen to allowuser interaction with the test instrument 12. The test instrument 12 mayalso include any number of additional input/output devices, interfacedevices, and/or other peripheral devices. For example, in someembodiments, the peripheral devices may include a touch screen, graphicscircuitry, keyboard, mouse, speaker system, network interface, and/orother input/output devices, interface devices, and/or peripheraldevices.

Referring now to FIG. 2, an exemplary circuitry 26 illustratingarchitecture of the transmitter 14 communicatively coupled to the RFduplex 18 is shown. The transmitter 14 includes one or moredigital-to-analog (DA) converters 27 that are configured to convertincoming RF signal samples in digital form into an analog RF signal. Theanalog RF signal may be modulated prior to being transmitted to the UUT24. Referring now to FIG. 3, an exemplary circuitry 28 illustratingarchitecture of the receiver 16 communicatively coupled to the RF duplex18 is shown. The receiver 16 includes one or more analog-to-digital (AD)converters 29 that are configured to convert analog RF output signalinto a digitized RF signal. The digitized RF signal may be demodulatedprior to being outputted to the controller 20. It should be appreciatedthat, in some embodiments, the transmitter 14 and the receiver 16 may becombined into a transceiver that both transmits and receives signals.

The controller 20 may be embodied as any type of computation or computerdevice capable of performing the functions described herein, including,without limitation, a computer, a server, a workstation, a laptopcomputer, a notebook computer, a tablet computer, a mobile computingdevice, a wearable computing device, a network appliance, a distributedcomputing system, a processor-based system, and/or a consumer electronicdevice. As shown in FIG. 4, the controller 20 illustratively includes aprocessor 30, a memory 32, an input/output (I/O) subsystem 34, acommunication subsystem 36, a field-programmable gate array (FPGA) 38,one or more data storages 40, and one or more peripheral devices 42. Itshould be appreciated that the controller 20 may include other oradditional components, such as those commonly found in a computer (e.g.,various input/output devices), in other embodiments. Additionally, insome embodiments, one or more of the illustrative components may beincorporated in, or otherwise form a portion of, another component. Forexample, the memory 32, or portions thereof, may be incorporated in theprocessor 30 in some embodiments.

The processor 30 may be embodied as any type of processor capable ofperforming the functions described herein. For example, the processor 30may be embodied as a single or multi-core processor(s), digital signalprocessor, microcontroller, or other processor or processing/controllingcircuit. Similarly, the memory 32 may be embodied as any type ofvolatile or non-volatile memory or data storage capable of performingthe functions described herein. In operation, the memory 32 may storevarious data and software used during operation of the test instrument12 such as operating systems, applications, programs, libraries, anddrivers. The memory 32 is communicatively coupled to the processor 30via the I/O subsystem 34, which may be embodied as circuitry and/orcomponents to facilitate input/output operations with the processor 30,the memory 32, and other components of the controller 20. For example,the I/O subsystem 34 may be embodied as, or otherwise include, memorycontroller hubs, input/output control hubs, firmware devices,communication links (i.e., point-to-point links, bus links, wires,cables, light guides, printed circuit board traces, etc.) and/or othercomponents and subsystems to facilitate the input/output operations. Insome embodiments, the I/O subsystem 34 may form a portion of asystem-on-a-chip (SoC) and be incorporated, along with the processor 30,the memory 32, and other components of the controller 20 (e.g.,peripheral devices 42), on a single integrated circuit chip.

The communication subsystem 36 may be embodied as any type ofcommunication circuit, device, or collection thereof, capable ofenabling communications between the controller 20 and the UUT 24. To doso, the communication subsystem 36 may be configured to use any one ormore communication technologies (e.g., wireless or wired communications)and associated protocols (e.g., Ethernet, Bluetooth®, Wi-Fi®, WiMAX,LTE, 5G, etc.) to effect such communication. The FPGA 38 may be embodiedas any type of circuit that includes an array of programmable logicblocks that is configurable to estimate and compensate IQ imbalance. Thedata storage 40 may be embodied as any type of device or devicesconfigured for short-term or long-term storage of data such as, forexample, memory devices and circuits, memory cards, hard disk drives,solid-state drives, or other data storage devices. In the illustrativeembodiment, the data storage 40 may store estimated complex filters forthe transmitter 14 and the receiver 16 at observed RF frequencies forcalibration.

Referring now to FIG. 5, an exemplary circuitry 50 illustratesarchitecture of the controller 20 having the FPGA 38, which includes aplurality of routing channels. For example, in the illustrativeembodiment, the FPGA 38 includes two routing channels 52 (i.e., ADCchannels) that are communicatively coupled to the receiver 16 and tworouting channels 54 (i.e., DAC channels) that are communicativelycoupled to the transmitter 14. The incoming ADC channels 52 areconnected to an IQ imbalance compensator 60, an amplitude correction 62,a resampling engine 64, and an acquisition (ACQ) engine 66. Similarly,the outgoing DAC channels 54 are connected to an IQ imbalancecompensator 68, an amplitude correction 70, a resampling engine 72, andan arbitration (ARB) engine 74. The ACQ engine 66 and the ARB engine 74are configured to communicate with a trigger routing matrix 76. Itshould be appreciated that, in some embodiments, each of IQ imbalancecompensators 60, 68 may be embodied as part of the transmitter 14 andthe receiver 16.

A. IQ Estimation and Compensation

As described above, IQ imbalance is mainly introduced by mixerimperfections and frequency differences between in-phase (I) andquadrature (Q) channels. This impairment produces degradation in theperformance of both communication systems and test equipment. The mainsteps of a method of estimating the complex filters for the transmitter(TX) 14 and the receiver (RX) 16 are summarized below. The followingdefinitions are useful to describe the method:

-   -   O₁ is the observation at frequency f when the complex sine at        frequency f is transmitted.    -   O₂ is the observation at frequency −f when the complex sine at        frequency f is transmitted.    -   O₃ is the observation at frequency −f when the complex sine at        frequency −f is transmitted.    -   O₄ is the observation at frequency f when the complex sine at        frequency −f is transmitted.    -   O_(n,Δ) is the O_(n) observation with an inherent amplitude        imbalance on the I channel of Δη.

1) Transmitter (TX) Imbalance Filter Compensation Estimation:

The TX imbalance may be extracted from one observation if the phase issynchronized. Since such synchronization is difficult to achieve in thehardware, a second signal with amplitude imbalance may also betransmitted. One approach for extracting the TX imbalance is describedin De Witt, J. J. (2011), Modelling, estimation and compensation ofimbalances in quadrature transceivers (dissertation). As a summary, theTX compensation filter at frequency f_(m) is given by:

$\begin{matrix}{{{{Q\left( f_{m} \right)} = \frac{{{\eta_{M}\left( f_{m} \right)}e^{j\;\psi}M^{({fm})}} - 1}{{{\eta_{M}\left( f_{m} \right)}e^{j\;\psi}M^{(f_{m})}} + 1}},{where}}{{\eta_{M}\left( f_{m} \right)} = {\frac{\sqrt{{\Delta\eta}\; x\; y}}{{\Delta\eta}\; y}}}{and}{{{\psi_{M}\left( f_{m} \right)} = {{\pm \arccos}{\frac{\left( {{\Delta\;\eta} - 1} \right)\left( {{\Delta\eta} + 1} \right)\left( {K_{2} - 1} \right)\left( {K_{1} - 1} \right)}{2\;\sqrt{{\Delta\eta}\; x\; y}}}}},{where}}{{x = {1 + K_{1} - K_{2} - {K_{2}K_{1}} + {{\Delta\eta}\; K_{1}} - {\Delta\eta} + {{\Delta\eta K}_{2}K_{1}} - {{\Delta\eta}\; K_{2}}}},{y = {{- {\Delta\eta}} - {{\Delta\eta}\; K_{1}} + {\Delta\;\eta\; K_{2}} + {\Delta\;\eta\; K_{2}K_{1}} + 1 - K_{1} + K_{2} - {K_{2}K_{1}}}},{K_{1} = \frac{o_{4}}{o_{1}}},{and}}{K_{2} = {\frac{o_{4,\Delta}}{o_{1,\Delta}}.}}} & (1)\end{matrix}$It should be noted in Equation (1) that the sign of angle ψ_(m) cannotbe resolved. In consequence, it may be necessary to test the twopossibilities and choose the value that produces the best, e.g.,highest, IQ imbalance image rejection.

2) Receiver (RX) Imbalance Filter Compensation Estimation:

One approach for computing the RX imbalance filter compensation isdescribed in De Witt (2011). The IQ imbalance filter compensation forthe receiver (RX) imbalance can be computed as:

$\begin{matrix}{{{P\left( {- f_{r}} \right)} = \frac{o_{i,{f\; r}}}{o_{d,{fr}}^{*}}},} & (2)\end{matrix}$where O_(i,f) _(r) is the observation O_(i) at frequency f_(r).

B. Fitting a Surface-Method of Least Squares

As described in greater detail below, the complex observations outlinedabove can be fitted to a polynomial function using a least or minimumsquares approach.

1) Problem Formulation:

Given N×M many points at positions (x_(i), y_(j)) in

² where i∈[1 . . . A]j∈[1 . . . M]. To obtain the polynomial functionp(x,y) of degree m_(x) in x and m_(y) in y that approximates the givenscalar values p_(i,j) at points (x_(i), y_(j)) using the least squaresapproximation approach according to the error functionJ_(LS)=Σ_(i,j)∥p(x_(i),y_(i))−p_(i,j)∥², a minimization of the error maybe such that:

$\begin{matrix}{{\min\limits_{p\; \in {\mathbb{R}}^{2}}{\sum\limits_{i,j}{{{p\left( {x_{i},y_{i}} \right)} - p_{i,j}}}^{2}}},} & (3)\end{matrix}$where p is a function in

² with k number of coefficients and can be written asp(x,y)=b(x,y)^(T) c=b(x,y)·c,  (4)where b(x,y) is the polynomial basis vector and c=[c₁, . . . ,c_(k)]^(T) is the vector of unknown coefficients to be minimized usingEquation (3).

At this point, note that the basis vectors in Equation (4) may becomevery large as m_(x) and m_(y) increase, which can make the systeminefficient to solve later on. In order to shorten the basis vector, xand y are considered to be not correlated and, therefore, allcross-terms of p(x,y) will be zero (0). Accordingly, k=m_(x)+m_(y)+1.For convenience, the x and y terms were grouped in decreasing order withthe constant term being last, such thatb(x,y)=[x ^(m) ^(x) , . . . ,x ¹ ,y ^(m) ^(y) , . . . ,y ¹,1]^(T).  (5)

2) Solution:

A system is constructed to minimize Equation (3) by using the method ofnormal equations:

$\begin{matrix}{{{\begin{bmatrix}{b^{T}\left( {x_{1},y_{1}} \right)} \\\vdots \\{b^{T}\left( {x_{i},y_{j}} \right)}\end{bmatrix}c} = \begin{bmatrix}p_{1,1} \\\vdots \\p_{i,j}\end{bmatrix}},} & (6)\end{matrix}$which can be readily solved asc=(B ^(T) B)⁻¹ B ^(T) p.  (7)

It should be noted that the Equation (6) requires the use of an inversefunction, and is, therefore, sometimes inefficient in practice. Solvingthe system of Equation (7) by computing a QR factorization via the Eigenlibrary's householder method achieves acceptable performance.

Earlier in the Equation (5), all cross-terms in the system were assumedto be zero in an effort to reduce the computational complexity of thesystem. This is useful in practice since, in some cases, m_(x) and m_(y)are found to be as high as 15 in order to produce a good fit—meaningthat several hundred cross-terms may need to be considered in thesecases. Nevertheless, the quality of the fitted surface may be improvedby introducing a predefined number of the total cross-terms. Choosing toallow cross-terms of order m_(xy) and fewer introduces

$k_{x\; y} = \frac{m_{x\; y}\left( {m_{xy} - 1} \right)}{2}$more terms to Equation (5), and the system can be solved in a similarfashion.

The quality of the fitted surface can also be improved by applying acenter-shift transformation to the input coordinates (x_(i), y_(i)).This is done by centering x and y around 0 and scaling them to unitstandard deviation, such that

$\hat{x} = \frac{x - \overset{\_}{x}}{\sigma_{x}}$ and${\hat{y} = \frac{y - \overset{\_}{y}}{\sigma_{y}}},$which is a process that improves the arithmetic properties of thesystem.

In order to estimate the IQ imbalance using the estimation methoddescribed above, a complex tone must be transmitted and received in thesame device. In addition, to estimate both the TX and RX imbalancesimultaneously, an offset between the TX and RX mixer is required. Sinceit is desired to estimate simultaneously several IQ imbalance errors toreduce the calibration time, several complex sines should be transmittedsimultaneously. Therefore, the transmitted signal should be as follows:s[n]=Σ_(m=0) ^(m=N) e ^(j2πmfn),where N is the total number of desired observations on the positivefrequencies (for the negative frequencies, should be transmitted).However, if too many sine waves are transmitted, a high peak to averageratio can be obtained. Since the signal is digitized to estimate the IQimbalance, precision can be lost at the AD converter. Therefore, all therequired sine waves should be split into different signals that aretransmitted at different times. Additionally, a fast Fourier transform(FFT) may be used in the receiver to estimate the IQ imbalance. Toensure that an FFT is as small as possible, an offset between thereceiver and the transmitter that is multiple of a bin size may be used,such that the signal and the images lie on an FFT bin.

Considering above features, the following K signals are defined:

$\begin{matrix}{{{s_{p}\lbrack n\rbrack} = {\sum\limits_{m = 0}^{m = {\lbrack\frac{N}{K}\rbrack}}e^{j\; 2\;\pi\;{f_{steps}{({p + {Km}})}}n}}},} & (8)\end{matrix}$where N is the total number of observations, K indicates the distance(Kf_(step), Hz) between tones for the multi-sine signal, p=[0, . . . ,K−1] indicates the initial frequency offset (pf_(step), Hz) of thecomplex sines to transmit,

${f_{steps} = \frac{Bw}{N}},$and Bw is the baseband bandwidth to be compensated.

Signal s_(p)(n) will be transmitted to estimate the IQ imbalance ofpositive frequencies and s_(p)*(n) to estimate the IQ imbalance ofnegative frequencies signals. It should be noted that these signals meetthe requirements that several complex sines are transmitted and that theoffset p is a multiple of the bin size. Furthermore, the frequencies ofthe signal are separated by a bin distance. Therefore, all of them willlie on the bins of the FFT because there is not extra frequency offset.It is important to note that, since the transmitter and receiver areconnected to the same clock, they will perfectly synchronize and thetransmitted complex sine waves and its IQ imbalance images lie exactlyon the FFT bin.

A second amplitude imbalanced signal is required to estimate thetransmitter imbalance compensation if phase synchronization is notguaranteed. Therefore, the following amplitude imbalanced signal istransmitted:

$\begin{matrix}{{S_{p,\Delta}\lbrack n\rbrack} = {{\sum\limits_{m = 0}^{m = {\lbrack\frac{N}{K}\rbrack}}{\Delta\;\eta\;{\cos\left( {2\pi\;{f\left( {p + {Km}} \right)}n} \right)}}} + {j\;{{\sin\left( {2\pi\;{f\left( {p + {Km}} \right)}n} \right)}.}}}} & (9)\end{matrix}$It is possible to find the frequency response of the compensation filterfor the receiver, {tilde over (H)}_(rx), and transmitter, {tilde over(H)}_(tx), by using the s_(p)[n] and s_(p,Δ)[n] signals and thealgorithm summarized in above.

The direct current (DC) offset introduced by the DAC affects theobservation and, thus, also affects the estimations of the compensationfilters. Accordingly, two special complex tones,s _(tone)[n]  (10)have to be sent at f_(steps) and −f_(steps). These tones are used toestimate the frequency response of the compensation filter at thebaseband frequencies f_(steps) and −f_(steps).

Defining s_(p) ^(f)[n] as the signal s_(p)[n] at the RF frequency f and{tilde over (H)}_(f)[k] as the baseband frequency compensation estimatedfilter at the RF frequency f, it is possible to define the matrix ofobservations, {tilde over (H)}, as a surface, B, such that:

$B = \begin{bmatrix}{{{\overset{\sim}{H}}_{0}\lbrack 0\rbrack},} & {{\overset{\sim}{H}}_{0}\lbrack 1\rbrack} & \ldots & {{\overset{\sim}{H}}_{0}\lbrack N\rbrack} \\{{{\overset{\sim}{H}}_{1}\lbrack 0\rbrack},} & {{{\overset{\sim}{H}}_{1}\lbrack 1\rbrack},} & \ldots & {{\overset{\sim}{H}}_{1}\lbrack N\rbrack} \\\vdots & \vdots & \vdots & \vdots \\{{{\overset{\sim}{H}}_{M}\lbrack 0\rbrack},} & {{{\overset{\sim}{H}}_{M}\lbrack 1\rbrack},} & \ldots & {{\overset{\sim}{H}}_{M}\lbrack N\rbrack}\end{bmatrix}$These observations generate a surface that can be approximated to apolynomial equation.

As described above, the complex observations can be fitted using theleast squares approach to a polynomial function of two variables,p(f_(RF), f_(BB)), where f_(RF) is the RF frequency and f_(BB) is thebaseband frequency. The transmitter and receiver polynomial functionapproximations are defined as p_(tx)(f_(RF), f_(BB)) and p_(rx)(f_(RF),f_(BB)), respectively.

There are two potential approaches to approximate the surface tocompensate. A first approach, a polynomial function that fits the entireRF-frequency- and BB-frequency-based surface. This approach has theadvantage that frequency tables for all the observations do not need tobe stored. In this case, only a polynomial function equation isrequired. However, since a big surface is fitted, a bad fit on certainfrequencies may be found. The second option is to fit the surface aroundthe RF frequency that is going to be compensated. In this case, tablesof observations must be stored but better performance is expected.

Once the polynomial function is estimated, a compensation filter at adesired frequency, F_(RF), can be calculated. The baseband frequencyresponse of the compensation filter is p(F_(RF), f_(BB)). Furthermore, avector of estimation values at some baseband frequencies, {right arrowover (f)}_(BB), can be calculated: {right arrow over (p)}(F_(RF), {rightarrow over (f)}_(BB)). It should be noted that the RF frequency and thevector of baseband frequencies, {right arrow over (f)}_(BB), can be oneof those of the observation frequencies or not.

The complex filter impulse response can be estimated by finding the setof filter taps that minimize the error (in the least squares sense)between the desired and generated signals. The following equation may beused to calculate the filter taps:

$\overset{->}{a} = {Q^{- 1}\overset{->}{b}}$ where$Q_{n,k} = {\sum\limits_{m = 0}^{M - 1}{w_{m}e^{{- j}\; 2\;\pi\;{f_{m}{({n + k - {2\rho}})}}}}}$and$b_{k} = {\sum\limits_{m = 0}^{M - 1}{w_{m}D_{m}e^{{- j}\; 2\;\pi\;{f_{m}{({k - \rho})}}}}}$and where {right arrow over (a)} is indicative of computed taps, w isindicative of weights, f is the frequency at which the observations aretaken, and D is the desired frequency response at the frequency (orfrequencies) f.

In other embodiments, the impulse response of the filter can beestimated using the window method. In this method, the impulse responseof a desired frequency response is derived by means of an inverse FFT.The length of the frequency response, {right arrow over (f)}_(BB),should be the same as that of the desired filter length, L, and equallyspaced on the band to compensate. The impulse response is defined as:{tilde over (h)} _(un) =

{{right arrow over (f)} _(BB)}{tilde over (h)}[n]=w[n]·{tilde over (h)} _(un) [n], 0≤n<Nwhere {tilde over (h)} is the estimated compensation filter and w is thedesired window. It should be noted that two different filters need to beestimated, one for the transmitter and another one for the receiver.

As described above, the algorithm for estimating and compensating thetransmitter (TX) and receiver (RX) IQ imbalance by the testing systemincludes transmitting complex multi-sine signals for desired RFfrequencies, estimating the frequency responses in the desired RFfrequencies, determining a polynomial fit for a region around a targetRF frequency to compensate based on the estimations, calculating, usingthe polynomial fit, a frequency response of the compensation filter, andcalculating complex impulse responses for a TX pre-compensation filterand RX post-compensation filter using the window method.

The test instrument 12 produces high IQ imbalance images in both thetransmitter 14 and receiver 16. FIG. 7 shows an exemplary graph 80 of acomplex multi-sine signal 82 transmitted at a predefined centerfrequency, e.g., 5.125 GHz, without TX pre-compensation. As illustratedby element 84, the TX imbalance impairments are as high as 30 decibelsrelative to the carrier (dBc), e.g., a difference between magnitudelevel 81 of the desired signal and magnitude level 83 of the imbalanceimpairments is as high as 30 dB.

The algorithms outlined above were used to estimate and compensate theIQ imbalance. In consequence, the signals s_(p)(n) and s_(p)*(n) weretransmitted and then received to generate the compensation filtersurfaces for the transmitter and receiver. Data from the test instrument12 was captured for RF frequencies between 1500 and 5600 MHz at 100-MHzsteps. The baseband bandwidth for each captured RF frequency was 250MHz. The transmitter and receiver surfaces 86, 92 are shown in FIGS. 8Aand 9A, respectively. As shown in FIGS. 8A and 9A, the observationscontained significant noise 87, 93. This noise is due mainly to RF spursand the thermal noise introduced at the transmitter and receiver. Sincethe compensation filter has to be calculated at any RF frequency ofinterest, these filters are fit using the least squares approach to amulti-variable polynomial function, as described above. This polynomialfunction created the surfaces 88, 94 shown in FIG. 8B and FIG. 9B. Itshould be noted that the surfaces 88, 94 do not contain the noise 87, 93of the observation surfaces 86, 92, respectively.

The first approach for estimating the compensation filters at a targetRF frequency was by fitting all the compensation filters at theobservation RF frequencies using the surface polynomial function 88described above. FIG. 8B shows a surface fit 88 for all the compensationfilters estimated based on observations captured in the transmittersurface 86 of FIG. 8A. To demonstrate that the complex compensationfilters derived from the polynomial function reduce the IQ imbalanceimages, a pre-compensated complex multi-sine signal was transmitted atsome different RF frequencies than those used during the observationprocess. This approach reduced significantly the IQ imbalance imagessince the polynomial function fit is very close to the originalmeasurements. Pre-compensating the TX signal with the compensationfilter estimated using the polynomial function suppresses the IQ imagesfor most of the RF frequencies to values lower than 55 dBc. However, insome cases images as high as 51 dBc were measured. Since images lowerthan 60 dBc are desired, a polynomial function that fits only a local(i.e., limited frequency range) surface was preferred. Example graphs90, 96 of FIGS. 8C and 9C illustrate local transmitter and receiversurfaces fit, respectively. Accordingly, storing the estimated complexfilters at the observed RF frequencies is required for calibration. Theparameters used to estimate and compensate the IQ imbalance for thelocal fit are summarized in Table 1 in FIG. 6.

A local fit of four different observed RF frequencies around the RFfrequency to compensate was used to generate the response shown in FIG.10A. Images as low as 58 dBc were measured at an external spectrumanalyzer. In order to reduce further these images, observations every 50MHz around the points to calibrate were captured. This reduced the TX IQimbalance image magnitude to values below 60 dBc. FIG. 10A shows anexemplary graph 98 illustrating the spectrum 100 of the images of the TXimbalance at 5.125 GHz. As illustrated, for example, by element 102, thehighest image is −61.7 dBc in a 200-MHz bandwidth, e.g., a differencebetween magnitude 116 of the desired signal and magnitude 118 of thehighest imbalance image is at least 60 dB. It should be noted that spurs104 at −80 MHz are not due to TX IQ imbalance impairments. This means animprovement of more than 30 dB for the image attenuation compared to theuncompensated TX images at the same frequency illustrated by element 84of FIG. 7. It is important to highlight that spurs at other frequenciesare below 65 dBc.

Although cross terms were used for the polynomial function fit, therewas not a measurable difference when compared to polynomial functionsthat only use independent variables between the RF frequency axis andthe baseband axis. This was expected because these parameters should beindependent.

FIGS. 10B and 10C show exemplary graphs 106, 108 of the output afterpre-compensation in the transmitter and post-compensation in thereceiver, respectively. The images after compensation in a basebandbandwidth of 160 MHz are below 60 dBc, e.g., illustrated by a relativedifference between zero (0) dB magnitude level and each of magnitudelevels 110 and 112 of FIGS. 10B and 10C, respectively. The 160 MHz isintroduced by the receiver. It should be noted that the spur 114 at −8.9MHz in FIG. 10C is not due to the IQ imbalance.

As discussed above, FIGS. 7-10C illustrate IQ imbalance rejection forpre-compensated and post-compensated signals in the test instrument 12.IQ images suppression of more than 60 dBc was achieved for bothtransmitter 14 and receiver 16. For example, to achieve that level ofsuppression, the baseband bandwidths required for the transmitter 14 andthe receiver 16 are 200 MHz and 160 MHz, respectively. These resultsshow improvements compared to the 52 dBc image suppression specified bythe M9381A PXIe Vector signal generator, which is available commerciallyfrom Keysight Technologies, Inc.

In one illustrative embodiment, the compensation process includes thefollowing steps:

-   -   1) Connect the receiver 16 and the transmitter 14;    -   2) Compensate the baseband filters using the pseudo-noise (PN)        sequence method. See, for example, Vergel, Julio and Roberts,        Jean. Study of different signals suitable to obtain the        frequency of an unknown stable linear time-invariant (LTI)        system, for an exemplary PN sequence method;    -   3) Compensate for direct current (DC) offset in the transmitter        and the receiver;    -   4) Set the transmitter RF, f_(tx), frequency to the desired        frequency to estimate the compensation and the receiver RF        frequency to f_(rx)=f_(tx)−2 MHz;    -   5) Transmit the s_(p)[n], s_(p)*[n], s_(p,Δ)[n], and s_(p,Δ)*[n]        signals. Save received signals;    -   6) Set the transmitter RF, f_(tx), frequency to the desired        frequency to estimate the compensation and the receiver RF        frequency to f_(rx)=f_(tx)−4 MHz;    -   7) Transmit complex sine and its conjugate (s_(tone)[n],        s_(tone)*[n]);    -   8) Use observations to estimate transmitter and receiver        compensation filters; and    -   9) Load filters into the FPGA and begin testing.

In some embodiments, a user may also verify that IQ imbalance images are60 dBc or lower after the filters are loaded into the FPGA.

Finally, a calibration process was introduced based on the algorithmsthat estimate and compensate the IQ imbalance. To determine the final IQimage suppression value, the test instrument 12 may further beconfigured to measure the IQ imbalance suppression if the number ofobservation in baseband (N) is increased (e.g., from 125 to 250),measure the IQ imbalance suppression if the RF interval is reduced(e.g., from 50 MHz to 25 MHz), measure the IQ imbalance for the newgenerators, inspect variations of IQ imbalance suppression withtemperature, and/or test IQ imbalance for more RF frequencies toguarantee at least 60 dB IQ image suppression.

C. Software Interfaces

Three sets of software interfaces are provided for C++ implementation ofthis algorithm: Complex Sine generator, TX Imbalance estimation andcompensation, and RX Imbalance estimation and compensation.

1) Complex Sine Generator—AlgGenMultiSineInterface:

In the illustrative embodiment, this interface generates the multi-sineand complex tone signals to be transmitted from the transmitter asspecified in at least Equations (8), (9), and (10). Using the values inTable 1 shown in FIG. 6, the steps to generate the required signals are:

-   -   1) Generate the multi-complex sine waves, e.g., as per Equation        (8), at frequencies [0, 3/125, . . . , 123/125], [0, −3/125, . .        . , −123/125], [1/125, 4/125, . . . , 124/125], [−1/125, −4/125,        . . . , −124/125], [2/125, 5/125, . . . , 122/125], [−2/125,        −5/125, . . . , −122/125] in two steps:        -   a) For each set of multi-complex sine:            -   i) Set the initial phase to zero for all the sine                frequencies in the multi-sine signal using the                SetInitialPhases method; and            -   ii) Call the operator ( ) where the output is going to                be generated and the normalized frequencies.        -   b) For each set of multi-complex sine waves generate the            imbalanced signals, e.g., as per Equation (9).    -   2) Generate the single tones

Exemplary pseudo-code for the sine wave generator is shown in Algorithm1.

Algorithm 1 Pseudo code to generate signals autogen=Ngmp::DSP::AlgGenMultiSineFast< float, std::complex<float >>::Create( ); k=0 for (k<K) { frequencySet=[k:3:125)//Create kset of frequencies; vectorInit=0 //Initialize phases to zerogen.SetInitialPhases(vectorInit); (*gen)(outputPos,frequencySet) //Positive Multi (*gen)(outputNeg,frequencySet) // Negative MultioutputImbPos = real(outputPos)+2*imag(outputPos); outputImbNeg =real(outputNeg)+2*imag(outputNeg); } frequencySet=[1](*gen)(outputTonePos,frequencySet) // Positive Tone(*gen)(outputToneNeg,−frequencySet) // Negative Tone //Create imbalancesignals for tones

2) TX Imbalance Estimation Interface:

In the illustrative embodiment, the transmitter imbalance has eight mainmethods that can be used to estimate and compensate the TX IQ imbalance.The TX and RX interface are defined by the AlgRxImbalanceDeWittInterfaceand AlgTxImbalanceDeWittInterface classes, respectively. The stepsinclude:

-   -   Set the number of complex sines that were transmitted (N).        Currently, the number of observations is N=250.    -   Set the frequency offset between the receiver and transmitter in        bins. Since the FFT size is the same as the number of        observations, the offset in bins is such that

$\frac{f_{offset}f_{sampling}}{N}.$since the sampling frequency is 250 MHz, the offset in bins should besuch that

$f_{bins} = {\frac{250f_{offset}}{N} = 1.}$

-   -   Set the imbalance for the required imbalance signal to be        transmitted. Currently, the imbalance is 2.    -   For each RF frequency        -   Create a structure of RxSignal type and fill it with the            received signals;        -   Compute the frequency response of the compensation filter            for the RF frequency; and        -   Store the frequency response in a matrix where the rows are            the RF frequencies and the columns are the frequency            responses.    -   Calculate the surface polynomial that compensates several RF        frequencies passing the matrix and the vector of RF frequencies.    -   Transmit the multi-tone signal using the two options for the        compensation filter to determine the sign of the imaginary part        of the compensation filter.

a) void SetNumberOfObservations (std::int32_t numberOfObservations):Method used to set the total number of transmitted tones at oneparticular RF frequency. For example, for measurements every 1 MHz in agiven baseband, 250 observations are required for a 250 MHz basebandbandwidth.

b) void SetFrequencyOffsetTxRxlnBins (std::int32_tfrequencyOffsetMultiTxRxToSet, std::int32_tfrequencyOffsetToneTxRxToSet): Indicates the distance in bins betweenthe RX and TX mixers for the multi-sine signal and the one-tone signal.A value of one should be set for a 1 MHz offset between RX and TXfrequencies and 250 observations in a 250 MHz bandwidth.

c) void SetAmplitudeImbalance (GeneralType amplitudeImbalanceToSet): TheSetAmplitudeImbalance method is used to set the amplitude imbalance ofthe signal with IQ imbalance required by the algorithm. Currently, twois used as the value of amplitude imbalance.

d) ViaviAvCommDSP::Signal <InputType> operator( ) (const RxSignals&rxSignals): This operator calculates the frequency response of thecomplex filter that compensates the TX IQ imbalance for a particular RFfrequency. RxSignal is a structure that includes all the signalsrequired to estimate the TX IQ imbalance compensation complex filter.

e) ViaviAvCommDSP::Signal <InputType> operator( )(ViaviAvCommDSP::ConstSignal <GeneralType> rfFrequencies,ViaviAvCommDSP::ConstMatrix2D <InputType> rxSignals): This operatorcalculates a polynomial equation surface for the observations in thematrix rxSignals at the RF frequencies rfFrequencies. The matrix andvector are related as follows:

$\left. \begin{bmatrix}f_{{rf}_{0}} \\f_{{rf}_{1}} \\\vdots \\f_{r\; f_{N - 1}}\end{bmatrix}\Rightarrow\begin{bmatrix}f_{B_{00}} & f_{B_{0\; 1}} & \ldots & f_{B_{0{({M - 1})}}} \\f_{B_{10}} & f_{B_{11}} & \ldots & f_{B_{1{({M - 1})}}} \\\vdots & \vdots & \vdots & \vdots \\f_{B_{{({N - 1})}0}} & f_{B_{{({N - 1})}1}} & \ldots & f_{B_{{({N - 1})}{({M - 1})}}}\end{bmatrix} \right.$For a given RF frequency, f_(rf) _(i) , the baseband compensation filterresponse is given by the row [f_(B) _(i0) f_(B) _(i1) . . . f_(B) _(i)_((M-1))]. The baseband compensation filter response, for a given RFfrequency, f_(rfi), should have been estimated using the operator insection VI-B4.

f) ViaviAvCommDSP::Signal <InputType> GetCompensatorImpulseResponse(ViaviAvCommDSP::Signal <InputType> frequencyResponse, conststd::uint32_t filterLength, const std::uint32_t polyOrder): Returns theimpulse response for a given frequency response.

g) GetCompensatorImpulseResponse (ViaviAvCommDSP::Signal <InputType>polynomialFunction, float frequencyRF, const std::uint32_tfilterLength): Interface to create an impulse response filter of filterlength that compensates the IQ imbalance at RF frequency frequencyRFusing the polynomial surface given by polynomialFunction.

h) ViaviAvCommDSP::ConstSignal <std::int16_t>DetermineSignImaginaryTx(const FrequencySide side,ViaviAvCommDSP::Signal <InputType> input, ViaviAvCommDSP::Signal<InputType> inputConj): Determine the sign of the imaginary part foreach tone frequency. Input and InputConj should be observations at thesame frequencies used to determine the IQ imbalance.

Exemplary pseudo-code to estimate the IQ imbalance compensation filteris shown in Algorithm 2.

Algorithm 2 Pseudo code to estimate TX imbalance autotxImb=Ngmp::DSP::AlgTxImbalanceDeWittNoPhase < float , std::complex<float > >::Create( ) txImb−>SetFrequencyOffsetTxRxInBins( 1 , 2 ); //Setdeltas between RX & TS txImb−>SetNumberOfObservations( 250 );//Set totalnumber of sines txImb−>SetAmplitudeImbalance( 2 );//Set Tx imbalance//Create signal structure with all the received tones at a given RFfrequency typename Ngmp::DSP::AlgTxImbalanceDeWittInterface<GeneralType,InputType>::RxSignals mySignals;mySignals.tone[AlgDeWittIf::positive]=tonePos;mySignals.tone[AlgDeWittIf::positiveImbalanced]=tonePosImb;mySignals.tone[AlgDeWittIf::negative]=toneNeg;mySignals.tone[AlgDeWittIf::negativeImbalanced]=toneNegImb;mySignals.multiTones[0][positive]=inputPos0;mySignals.multiTones[0][positiveImbalanced]=inputPosImb0;mySignals.multiTones[0][negative]=inputNeg0;mySignals.multiTones[0][negativeImbalanced]=inputNegImb0;mySignals.multiTones[1][positive]=inputPos1;mySignals.multiTones[1][positiveImbalanced]=inputPosImb1;mySignals.multiTones[1][negative]=inputNeg1;mySignals.multiTones[1][negativeImbalanced]=inputNegImb1;mySignals.multiTones[2][positive]=inputPos2;mySignals.multiTones[2][positiveImbalanced]=inputPosImb2;mySignals.multiTones[2][negative]=inputNeg2;mySignals.multiTones[2][negativeImbalanced]=inputNegImb2; freqTx =(*objectToTest)( mySignals ); //EstimateFrequency response ofcompensator //Get filter response but fitting first to a polynomialfunction. const std::uint32_t polyOrder = 8; // Set order conststd::uint32_t desiredFilterLength = 31; autofreqTxImpulse=txImb−>GetCompensatorImpulseResponse( freqTx ,desiredFilterLength , polyOrder );

3) RX Imbalance Estimation Interface:

In the illustrative embodiment, the receiver (RX) interface has sixmethods to estimate and compensate the RX imbalance.

a) void SetNumberOfObservations(const std::int32_tnumberOfObservationsToSet): Method used to set the total number oftransmitted tones for one particular RF frequency. For example, formeasurements every 1 MHz in a given baseband, 250 observations arerequired for a 250 MHz baseband bandwidth.

b) void SetFrequencyOffsetTxRxlnBins(std::int32_tfrequencyOffsetMultiTxRxToSet, std::int32_tfrequencyOffsetToneTxRxToSet): Indicates the distance in bins betweenthe RX and TX mixers for the multi-sine signal and the one-tone signal.A value of one should be set for a 1 MHz offset between Rx and TXfrequencies and 250 observations in a 250 MHz bandwidth.

c) ViaviAvCommDSP::Signal <InputType> operator( ) (const RxSignals&rxSignals): This operator calculates the frequency response of thecomplex filter that compensates the Rx IQ imbalance for a particular RFfrequency. RxSignal is a structure that includes all the signalsrequired to estimate the Rx IQ imbalance compensation complex filter.The output of this operator, compensation filter at a particular RF,will be used in subsequent stages.

d) ViaviAvCommDSP::Signal <InputType> operator ( )(ViaviAvCommDSP::ConstSignal <GeneralType> rfFrequencies, ViaviAvCommDSP::ConstMatrix2D <InputType> rxSignals): This operator calculates apolynomial equation surface for the observations in the matrix rxSignalsat the RF frequencies rfFrequencies. The matrix and vector are relatedas follows:

$\left. \begin{bmatrix}f_{{rf}_{0}} \\f_{{rf}_{1}} \\\vdots \\f_{r\; f_{N - 1}}\end{bmatrix}\Rightarrow{\begin{bmatrix}f_{B_{00}} & f_{B_{0\; 1}} & \ldots & f_{B_{0{({M - 1})}}} \\f_{B_{10}} & f_{B_{11}} & \ldots & f_{B_{1{({M - 1})}}} \\\vdots & \vdots & \vdots & \vdots \\f_{B_{{({N - 1})}0}} & f_{B_{{({N - 1})}1}} & \ldots & f_{B_{{({N - 1})}{({M - 1})}}}\end{bmatrix}.} \right.$

For a given RF frequency, f_(rf) _(i) , the baseband compensation filterresponse is given by the row [f_(BB) _(i0) f_(BB) _(i1) . . . f_(BB)_(i(M-1)) ]. The baseband compensation filter response, for a given RFfrequency, f_(rf) _(i) , should have been estimated using the operatorin section VI-C3.

e) ViaviAvCommDSP::Signal <InputType> GetCompensatorImpulseResponse(ViaviAvCommDSP::Signal <InputType> frequencyResponse, conststd::uint32_t filterLength, const std::uint32_t polyOrder): Returns theimpulse response for a given frequency response.

f) GetCompensatorImpulseResponse(ViaviAvCommDSP::Signal <InputType>polynomialFunction, float frequencyRF, const std::uint32_tfilterLength): Interface to create an impulse response filter of lengthfilter length that compensates the IQ imbalance at RF frequency usingthe polynomial surface given by polynomialFunction.

While the disclosure has been illustrated and described in detail in thedrawings and foregoing description, such an illustration and descriptionis to be considered as exemplary and not restrictive in character, itbeing understood that only illustrative embodiments have been shown anddescribed and that all changes and modifications that come within thespirit of the disclosure are desired to be protected.

There are a plurality of advantages of the present disclosure arisingfrom the various features of the method, apparatus, and system describedherein. It will be noted that alternative embodiments of the method,apparatus, and system of the present disclosure may not include all ofthe features described yet still benefit from at least some of theadvantages of such features. Those of ordinary skill in the art mayreadily devise their own implementations of the method, apparatus, andsystem that incorporate one or more of the features of the presentinvention and fall within the spirit and scope of the present disclosureas defined by the appended claims.

The invention claimed is:
 1. A test instrument comprising: a transmitterconfigured to transmit signals to a unit under test; a receiverconfigured to receive signals from the unit under test; and a controllerconfigured to: generate a transmitter compensation filter at least by(i) transmitting, with the transmitter, complex multi-sine signals overa first plurality of observed frequencies within a predeterminedbaseband frequency range, (ii) estimating a first plurality of frequencyresponses that compensate for in-phase and quadrature (IQ) imbalance atthe first plurality of observed frequencies within the predeterminedbaseband frequency range, and (iii) determining, using the firstplurality of frequency responses, a transmitter polynomial surface, andcompensate, using the transmitter compensation filter, at least one ofthe signals to be transmitted by the transmitter to reduce IQ imbalancein the transmitted signals, including using the transmitter polynomialsurface to calculate a frequency response that reduces the IQ imbalancein the transmitted signals.
 2. The test instrument of claim 1, whereinthe controller is further configured to: generate a receivercompensation filter at least by (i) receiving, with the receiver,complex multi-sine signals over a second plurality of observedfrequencies within the predetermined baseband frequency range, (ii)estimating a second plurality of frequency responses that compensate forIQ imbalance at the second plurality of observed frequencies within thepredetermined baseband frequency range, and (iii) determining, using thesecond plurality of frequency responses, a receiver polynomial surface,and compensate, using the receiver compensation filter, at least one ofthe signals to be received by the receiver to reduce IQ imbalance in thereceived signals, including using the receiver polynomial surface tocalculate a frequency response that reduces the IQ imbalance in thereceived signals.
 3. The test instrument of claim 2, wherein to generatethe transmitter compensation filter further comprises transmitting anamplitude imbalance signal in response to phases of the transmitter andthe receiver not being synchronized.
 4. The test instrument of claim 2,wherein the controller is configured to generate the transmittercompensation filter and the receiver compensation filter at the sametime by offsetting respective operating frequencies of a mixer of thetransmitter and a mixer of the receiver during the transmitting andreceiving of the complex multi-sine signals.
 5. The test instrument ofclaim 4, wherein the controller is configured to offset the respectiveoperating frequencies by a multiple of a bin size of a fast Fouriertransform (FFT) block, such that each of complex sines and images of theIQ imbalance lie on a bin of the FFT block.
 6. The test instrument ofclaim 1, wherein the first plurality of observed frequencies comprises(i) a first observation at a positive predefined frequency when acomplex sine signal is transmitted at one of the positive predefinedfrequency and a complex conjugate of the positive predefined frequencyand (ii) a second observation at a negative predefined frequency when acomplex sine signal is transmitted at one of the negative predefinedfrequency and a complex conjugate of the negative predefined frequency.7. The test instrument of claim 1, wherein the controller is configuredto transmit the complex multi-sine signals by transmitting separatecomplex sine signals at different times.
 8. The test instrument of claim1, wherein the transmitter polynomial surface includes at least onefrequency outside the predetermined baseband frequency range.
 9. Thetest instrument of claim 1, wherein the IQ imbalance includes IQimbalance images, and wherein the controller is configured to reduce theIQ imbalance images to be at least 60 decibels (dB) below the signals tobe transmitted by the transmitter.
 10. The test instrument of claim 1,wherein the controller is configured to determine the transmitterpolynomial surface by approximating a multi-variable polynomial functionusing a least squares approach.
 11. The test instrument of claim 2,wherein the controller includes a transmitter IQ imbalance compensator,a receiver IQ imbalance compensator, a transmitter amplitude correctionengine, a receiver amplitude correction engine, a transmitter resamplingengine, a receiver resampling engine, an acquisition engine, anarbitration engine, and a trigger routing matrix, wherein theacquisition and arbitration engines are both in communication with thetrigger routing matrix.
 12. A method comprising: generating atransmitter compensation filter, wherein generating the transmittercompensation filter comprises the steps of (i) transmitting, with atransmitter, complex multi-sine signals over a first plurality ofobserved frequencies within a predetermined baseband frequency range,(ii) estimating a first plurality of frequency responses that compensatefor in-phase and quadrature (IQ) imbalance at the first plurality ofobserved frequencies within the predetermined baseband frequency range,and (iii) determining, using the first plurality of frequency responses,a transmitter polynomial surface.
 13. The method of claim 12, furthercomprising: compensating, using the transmitter compensation filter, asignal transmitted by the transmitter to reduce IQ imbalance in thetransmitted signal, including using the transmitter polynomial surfaceto calculate a frequency response that reduces the IQ imbalance in thetransmitted signal.
 14. The method of claim 12, further comprising:generating a receiver compensation filter, wherein generating thereceiver compensation filter comprises the steps of (i) receiving, witha receiver, complex multi-sine signals over a second plurality ofobserved frequencies within the predetermined baseband frequency range,(ii) estimating a second plurality of frequency responses thatcompensate for IQ imbalance at the second plurality of observedfrequencies within the predetermined baseband frequency range, and (iii)determining, using the second plurality of frequency responses, areceiver polynomial surface.
 15. The method of claim 14, furthercomprising: compensating, using the transmitter compensation filter, asignal transmitted by the transmitter to reduce IQ imbalance in thetransmitted signal, including using the transmitter polynomial surfaceto calculate a frequency response that reduces the IQ imbalance in thetransmitted signal; and compensating, using the receiver compensationfilter, a signal received by the receiver to reduce IQ imbalance in thereceived signal, including using the receiver polynomial surface tocalculate a frequency response that reduces the IQ imbalance in thereceived signal.
 16. The method of claim 14, wherein the transmittercompensation filter and the receiver compensation filter are generatedat the same time by offsetting respective operating frequencies of amixer of the transmitter and a mixer of the receiver during thetransmitting and receiving of the complex multi-sine signals.
 17. Themethod of claim 12, wherein the first plurality of observed frequenciescomprises (i) a first observation at a positive predefined frequencywhen a complex sine signal is transmitted at one of the positivepredefined frequency and a complex conjugate of the positive predefinedfrequency and (ii) a second observation at a negative predefinedfrequency when a complex sine signal is transmitted at one of thenegative predefined frequency and a complex conjugate of the negativepredefined frequency.
 18. A test instrument comprising: a transmitterconfigured to transmit signals to a unit under test; a receiverconfigured to receive signals from the unit under test; and a controllerconfigured to reduce in-phase and quadrature (IQ) imbalance of thesignals to be transmitted and received, at least by: transmittingcomplex multi-sine signals over a first plurality of observedfrequencies within a predetermined baseband frequency range, receivingthe transmitted multi-sine signals over a second plurality of observedfrequencies within the predetermined baseband frequency range, thesecond plurality of observed frequencies being offset from the firstplurality of observed frequencies, estimating corresponding sets offrequency responses for the transmitted and received signals thatcompensate for IQ imbalance at the first and second pluralities ofobserved frequencies within the predetermined baseband frequency range,determining, using the sets of frequency responses, polynomial surfaces,approximating each of the polynomial surfaces with a corresponding oneof transmitter and receiver multi-variable polynomials, and reducing theIQ imbalance of the signals to be transmitted and received based onfrequency responses calculated using the transmitter and receivermulti-variable polynomials.
 19. The test instrument of claim 18, whereinthe controller is configured to reduce the IQ imbalance such that IQimbalance images are at least 60 decibels (dB) below at least one of thetransmitted and received signals.
 20. The test instrument of claim 19,wherein the controller is configured to reduce the IQ imbalance forfrequencies covering a 6-GHz bandwidth.